BiG-VAMP: The Bilinear Generalized Vector Approximate Message Algorithm

We introduce the bilinear generalized vector approximate message passing (BiG-VAMP) algorithm which jointly recovers rank-deficient matrices from their noisy product through a probabilistic observation model. BiG-VAMP provides computationally efficient approximate implementations of both max-sum and sum-product loopy belief propagation (BP). We show how the proposed BiG-VAMP recovers different types of structured matrices and overcomes the fundamental limitations of other state-of-the-art techniques to the bilinear recovery problem, such as BiG-AMP, BAd-VAMP, and LowRAMP. In essence, BiG- VAMP applies to a broader class of practical applications that involve a general form of structured matrices. Numerical results on applications such as dictionary learning and real- valued/quantized matrix completion demonstrate the effectiveness of the proposed BiG-VAMP algorithm and its superiority over state-of-the-art techniques.